If you know about classical modular forms, and you want to understand what they have to do with tmf, it is good to contemplate something more classical: the "derived functors of modular forms". Modular forms of weight n are H<sup>0</sup>(M, ω<sup>n</sup>), where ω is a certain line bundle over M=compactified moduli stack of elliptic curves. There is non-trivial cohomology in H<sup>s</sup>(M,ω<sup>n</sup>) for s>0. This comes in two flavors: \* free abelian group summands in H<sup>1</sup>(M,ω<sup>n</sup>) for n≤-10. These correspond by a kind of "Serre duality" to the usual modular forms in H<sup>0</sup>(M,ω<sup>-n-10</sup>). \* finite abelian groups (killed by multiplication by 24) in H<sup>s</sup>(M,ω<sup>n</sup>) for arbitrarily large s. There is a spectral sequence H<sup>s</sup>(M,ω<sup>t/2</sup>) ==> π<sub>t-s</sub> Tmf, ![alt text][1] and the edge of the spectral sequence gives a map π<sub>2n</sub>Tmf -> H<sup>0</sup>(M,ω<sup>n</sup>). This is essentially the map Chris describes in his answer. Slightly confusingly, the gadget I've called Tmf (which is the global sections of a sheaf of spectra over M) is *not* the same as TMF (the 576-periodic guy, which is sections on M'=stack of *smooth* elliptic curves), or tmf (the connective cover of Tmf, which I don't believe is known to be sections on anything). I recommend Goerss's <a href="http://arxiv.org/abs/0910.5130">Bourbaki talk</a> for learning more about this, especially section 4.6. [1]: http://math.mit.edu/conferences/talbot/2007/tmfproc/EllipticSpectralSequence.pdf